# Copyright © 2009 Pearson Education, Inc. Chapter 35 Diffraction and Polarization.

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ConcepTest 35.1aDiffraction I The diffraction pattern below arises from a single slit. If we would like to sharpen the pattern, i.e., make the central bright spot narrower, what should we do to the slit width? 1) narrow the slit 2) widen the slit 3) enlarge the screen 4) close off the slit

The angle at which one finds the first minimum is: The central bright spot can be narrowed by having a smaller angle. This in turn is accomplished by widening the slit. ConcepTest 35.1aDiffraction I d    sin  =  d The diffraction pattern below arises from a single slit. If we would like to sharpen the pattern, i.e., make the central bright spot narrower, what should we do to the slit width? 1) narrow the slit 2) widen the slit 3) enlarge the screen 4) close off the slit

ConcepTest 35.1bDiffraction II Blue light of wavelength passes through a single slit of width d and forms a diffraction pattern on a screen. If the blue light is replaced by red light of wavelength 2, the original diffraction pattern can be reproduced if the slit width is changed to: 1) d/4 2) d/2 3) no change needed 4) 2d 5) 4d

ConcepTest 35.1bDiffraction II d    d sin  = m (minima)  2,d  2dfor sin  to remain unchanged If  2, then we must have d  2d for sin  to remain unchanged (and thus give the same diffraction pattern). Blue light of wavelength passes through a single slit of width d and forms a diffraction pattern on a screen. If the blue light is replaced by red light of wavelength 2, the original diffraction pattern can be reproduced if the slit width is changed to: 1) d/4 2) d/2 3) no change needed 4) 2d 5) 4d

ConcepTest 35.2Diffraction Disk 1) darker than the rest of the shadow 2) a bright spot 3) bright or dark, depending on the wavelength 4) bright or dark, depending on the distance to the screen Imagine holding a circular disk in a beam of monochromatic light. If diffraction occurs at the edge of the disk, the center of the shadow is

interfere constructively in phasesame distance By symmetry, all of the waves coming from the edge of the disk interfere constructively in the middle because they are all in phase and they all travel the same distance to the screen. ConcepTest 35.2Diffraction Disk Imagine holding a circular disk in a beam of monochromatic light. If diffraction occurs at the edge of the disk, the center of the shadow is: Follow-up: Follow-up: What if the disk is oval and not circular? 1) darker than the rest of the shadow 2) a bright spot 3) bright or dark, depending on the wavelength 4) bright or dark, depending on the distance to the screen

Copyright © 2009 Pearson Education, Inc. Resolution is the distance at which a lens can barely distinguish two separate objects. Resolution is limited by aberrations and by diffraction. Aberrations can be minimized, but diffraction is unavoidable; it is due to the size of the lens compared to the wavelength of the light. 35-4 Limits of Resolution; Circular Apertures

Copyright © 2009 Pearson Education, Inc. For a circular aperture of diameter D, the central maximum has an angular width: 35-4 Limits of Resolution; Circular Apertures

Copyright © 2009 Pearson Education, Inc. The Rayleigh criterion states that two images are just resolvable when the center of one peak is over the first minimum of the other. 35-4 Limits of Resolution; Circular Apertures

Copyright © 2009 Pearson Education, Inc. 35-4 Limits of Resolution; Circular Apertures Example 35-5: Hubble Space Telescope. The Hubble Space Telescope (HST) is a reflecting telescope that was placed in orbit above the Earth’s atmosphere, so its resolution would not be limited by turbulence in the atmosphere. Its objective diameter is 2.4 m. For visible light, say λ = 550 nm, estimate the improvement in resolution the Hubble offers over Earth-bound telescopes, which are limited in resolution by movement of the Earth’s atmosphere to about half an arc second. (Each degree is divided into 60 minutes each containing 60 seconds, so 1° = 3600 arc seconds.)

Copyright © 2009 Pearson Education, Inc. 35-4 Limits of Resolution; Circular Apertures Example 35-6: Eye resolution. You are in an airplane at an altitude of 10,000 m. If you look down at the ground, estimate the minimum separation s between objects that you could distinguish. Could you count cars in a parking lot? Consider only diffraction, and assume your pupil is about 3.0 mm in diameter and λ = 550 nm.

Copyright © 2009 Pearson Education, Inc. For telescopes, the resolution limit is as we have defined it: 35-5 Resolution of Telescopes and Microscopes; the λ Limit For microscopes, assuming the object is at the focal point, the resolving power is given by

Copyright © 2009 Pearson Education, Inc. Typically, the focal length of a microscope lens is half its diameter, which shows that it is not possible to resolve details smaller than the wavelength being used: 35-5 Resolution of Telescopes and Microscopes; the λ Limit

Copyright © 2009 Pearson Education, Inc. A diffraction grating consists of a large number of equally spaced narrow slits or lines. A transmission grating has slits, while a reflection grating has lines that reflect light. The more lines or slits there are, the narrower the peaks. 35-7 Diffraction Grating

Copyright © 2009 Pearson Education, Inc. The maxima of the diffraction pattern are defined by 35-7 Diffraction Grating

Copyright © 2009 Pearson Education, Inc. 35-7 Diffraction Grating Example 35-8: Diffraction grating: lines. Determine the angular positions of the first- and second-order maxima for light of wavelength 400 nm and 700 nm incident on a grating containing 10,000 lines/cm.

Copyright © 2009 Pearson Education, Inc. 35-7 Diffraction Grating Example 35-9: Spectra overlap. White light containing wavelengths from 400 nm to 750 nm strikes a grating containing 4000 lines/cm. Show that the blue at λ = 450 nm of the third-order spectrum overlaps the red at 700 nm of the second order.

Copyright © 2009 Pearson Education, Inc. A spectrometer makes accurate measurements of wavelengths using a diffraction grating or prism. 35-8 The Spectrometer and Spectroscopy

Copyright © 2009 Pearson Education, Inc. The wavelength can be determined to high accuracy by measuring the angle at which the light is diffracted: 35-8 The Spectrometer and Spectroscopy

Copyright © 2009 Pearson Education, Inc. 35-8 The Spectrometer and Spectroscopy Atoms and molecules can be identified when they are in a thin gas through their characteristic emission lines.

Copyright © 2009 Pearson Education, Inc. 35-8 The Spectrometer and Spectroscopy Example 35-11: Hydrogen spectrum. Light emitted by hot hydrogen gas is observed with a spectroscope using a diffraction grating having 1.00 x 10 4 lines/cm. The spectral lines nearest to the center (0°) are a violet line at 24.2°, a blue line at 25.7°, a blue-green line at 29.1°, and a red line at 41.0° from the center. What are the wavelengths of these spectral lines of hydrogen?

Copyright © 2009 Pearson Education, Inc. 35-9 Peak Widths and Resolving Power for a Diffraction Grating As the number of slits becomes large, the width of the central maximum becomes very narrow: The resolving power of a diffraction grating is the minimum difference between wavelengths that can be distinguished:

Copyright © 2009 Pearson Education, Inc. 35-9 Peak Widths and Resolving Power for a Diffraction Grating Example 35-12: Resolving two close lines. Yellow sodium light, which consists of two wavelengths, λ 1 = 589.00 nm and λ 2 = 589.59 nm, falls on a diffraction grating. Determine (a) the maximum order m that will be present for sodium light, and (b) the width of grating necessary to resolve the two sodium lines.

Copyright © 2009 Pearson Education, Inc. The wavelengths of X-rays are very short. Diffraction experiments are impossible to do with conventional diffraction gratings. Crystals have spacing between their layers that is ideal for diffracting X-rays. 35-10 X-Rays and X-Ray Diffraction

Copyright © 2009 Pearson Education, Inc. X-ray diffraction is now used to study the internal structure of crystals; this is how the helical structure of DNA was determined. 35-10 X-Rays and X-Ray Diffraction

Copyright © 2009 Pearson Education, Inc. Light bends around obstacles and openings in its path, yielding diffraction patterns. Light passing through a narrow slit will produce a central bright maximum of width Minima occur at Summary of Chapter 35

Copyright © 2009 Pearson Education, Inc. Summary of Chapter 35 Diffraction limits the resolution of images. Diffraction grating has many parallel slits or lines; peaks of constructive interference are given by Polarized light has its electric field vectors in a single plane.

Copyright © 2009 Pearson Education, Inc. The intensity of plane-polarized light is reduced after it passes through another polarizer: Light can also be polarized by reflection; it is completely polarized when the reflection angle is the polarization angle: Summary of Chapter 35

Copyright © 2009 Pearson Education, Inc. Final Exam Thursday, 9:00 AM in this room Forty questions Two 8½” X 11” sheets, both sides, for equations Calculator #2 pencil Blank scratch paper Lucky talisman (NOT cell phone!) Strategy tip – if you’re not sure of an answer, eliminate those you KNOW are wrong, then guess. There’s no penalty for incorrect answers. GOOD LUCK!